Analysis of distrubances in large interconnected power systems

Analysis of Disturbances in Large Interconnected Power Systems by Mr.. Because the method relies on data from power systems in normal operation, the modal disturbances are random.. Keyw

By

Mr Richard Andrew Wiltshire

Bachelor of Engineering (Electrical and Computer Engineering)

1st Class Honours

Doctor of Philosophy

Centre of Energy and Resource Management

School of Engineering Systems Faculty of Built, Environment and Engineering

Queensland University of Technology

Analysis of Disturbances in Large Interconnected Power

Systems

by Mr Richard Andrew Wiltshire

Principal Supervisor: Associate Professor Peter O’Shea

School of Engineering Systems Faculty of Built, Environment and Engineering

Queensland University of Technology Associate Supervisor: Professor Gerard Ledwich

School of Engineering Systems Faculty of Built, Environment and Engineering

Queensland University of Technology Associate Supervisor: Dr Edward Palmer

School of Engineering Systems Faculty of Built, Environment and Engineering

Queensland University of Technology

Abstract

World economies increasingly demand reliable and economical power supply and distribution To achieve this aim the majority of power systems are becoming interconnected, with several power utilities supplying the one large network One problem that occurs in a large interconnected power system is the regular occurrence of system disturbances which can result in the creation of intra-area oscillating modes These modes can be regarded as the transient responses of the power system to excitation, which are generally characterised as decaying sinusoids For a power system operating ideally these transient responses

would ideally would have a “ring-down” time of 10-15 seconds

Sometimes equipment failures disturb the ideal operation of power systems and oscillating modes with ring-down times greater than 15 seconds arise The larger settling times associated with such “poorly damped” modes cause substantial power flows between generation nodes, resulting in significant physical stresses on the power distribution system

If these modes are not just poorly damped but “negatively damped”, catastrophic failures of the system can occur

To ensure system stability and security of large power systems, the potentially dangerous oscillating modes generated from disturbances (such as equipment failure) must be quickly identified The power utility must then apply appropriate damping control strategies

In power system monitoring there exist two facets of critical interest The

first is the estimation of modal parameters for a power system in normal, stable, operation The second is the rapid detection of any substantial

changes to this normal, stable operation (because of equipment breakdown for example) Most work to date has concentrated on the first

of these two facets, i.e on modal parameter estimation Numerous modal parameter estimation techniques have been proposed and implemented, but all have limitations [1-13] One of the key limitations of all existing parameter estimation methods is the fact that they require very long data records to provide accurate parameter estimates This is a particularly significant problem after a sudden detrimental change in damping One simply cannot afford to wait long enough to collect the large amounts of data required for existing parameter estimators Motivated by this gap in the current body of knowledge and practice, the research reported in this thesis focuses heavily on rapid detection of changes (i.e on the second facet mentioned above)

This thesis reports on a number of new algorithms which can rapidly flag whether or not there has been a detrimental change to a stable operating system It will be seen that the new algorithms enable sudden modal changes to be detected within quite short time frames (typically about 1 minute), using data from power systems in normal operation

The new methods reported in this thesis are summarised below

The Energy Based Detector (EBD): The rationale for this method is that

the modal disturbance energy is greater for lightly damped modes than it

is for heavily damped modes (because the latter decay more rapidly) Sudden changes in modal energy, then, imply sudden changes in modal damping Because the method relies on data from power systems in normal operation, the modal disturbances are random Accordingly, the disturbance energy is modelled as a random process (with the parameters

of the model being determined from the power system under consideration) A threshold is then set based on the statistical model The energy method is very simple to implement and is computationally efficient It is, however, only able to determine whether or not a sudden modal deterioration has occurred; it cannot identify which mode has deteriorated For this reason the method is particularly well suited to smaller interconnected power systems that involve only a single mode

Optimal Individual Mode Detector (OIMD): As discussed in the previous

paragraph, the energy detector can only determine whether or not a change has occurred; it cannot flag which mode is responsible for the deterioration The OIMD seeks to address this shortcoming It uses optimal detection theory to test for sudden changes in individual modes

In practice, one can have an OIMD operating for all modes within a system, so that changes in any of the modes can be detected Like the energy detector, the OIMD is based on a statistical model and a subsequently derived threshold test

The Kalman Innovation Detector (KID): This detector is an alternative to

the OIMD Unlike the OIMD, however, it does not explicitly monitor individual modes Rather it relies on a key property of a Kalman filter, namely that the Kalman innovation (the difference between the estimated and observed outputs) is white as long as the Kalman filter model is valid

A Kalman filter model is set to represent a particular power system If some event in the power system (such as equipment failure) causes a

sudden change to the power system, the Kalman model will no longer be valid and the innovation will no longer be white Furthermore, if there is a detrimental system change, the innovation spectrum will display strong peaks in the spectrum at frequency locations associated with changes Hence the innovation spectrum can be monitored to both set-off an

“alarm” when a change occurs and to identify which modal frequency has given rise to the change The threshold for alarming is based on the simple Chi-Squared PDF for a normalised white noise spectrum [14, 15] While the method can identify the mode which has deteriorated, it does not necessarily indicate whether there has been a frequency or damping change The PPM discussed next can monitor frequency changes and so can provide some discrimination in this regard

The Polynomial Phase Method (PPM): In [16] the cubic phase (CP)

function was introduced as a tool for revealing frequency related spectral changes This thesis extends the cubic phase function to a generalised class of polynomial phase functions which can reveal frequency related spectral changes in power systems A statistical analysis of the technique

is performed When applied to power system analysis, the PPM can provide knowledge of sudden shifts in frequency through both the new frequency estimate and the polynomial phase coefficient information This knowledge can be then cross-referenced with other detection methods to provide improved detection benchmarks

Keywords

Power System Monitoring, Interconnected Power Systems, Power System Disturbances, Power System Stability, Signal Processing, Optimal Detection Theory, Stochastic System Analysis, Kalman Filtering, Poly- Phase Signal Analysis

Declaration

I hereby certify that the work embodied in this thesis is the result of original research and has not been submitted for a higher degree

at any other University or Institution

Richard Andrew Wiltshire

10 July 2007

Table of Contents

Abstract iii

Keywords vi

Declaration vii

Table of Contents ix

Table of Figures xv

List of Tables xxi

Acknowledgements xxiii

Dedication xxv

Glossary xxvii

Chapter 1 29

1 Introduction 29

1.1 The Analysis of Large Interconnected Power Systems 29

1.2 The Monitoring of Australia's Large Interconnected Power System ……… 30

1.3 The use of Externally Sourced Simulated Data for Algorithm Verification 31

1.4 Review of Existing Modal Estimation Methods 33

1.4.1 Single Isolated Disturbance 33

1.4.1.1 Eigenanalysis of Disturbance Modes 33

1.4.1.2 Spectral Analysis using Prony’s Method 34

1.4.1.3 The Sliding Window Derivation 36

1.4.2 Continuous Random Disturbances 38

1.4.2.1 Autocorrelation Methods 38

1.4.2.2 Review of Kalman Filter Innovation Strategies 39

1.5 Review of Frequency Estimation Methods 40

1.5.1 Polynomial-Phase Estimation Methods 41

1.6 Conclusion 42

1.7 Organisation of the remainder of the thesis 43

Chapter 2 45

2 Rapid Detection of Deteriorating Modal Damping 45

2.1 Introduction 45

2.2 The Power System Model in the Quiescent State 46

2.3 The Power System Statistical Characterisation 47

2.4 PDF Verification 50

2.5 Setting the Threshold for Alarm 52

2.6 Simulated Results 52

2.7 Validation of Method using MudpackScripts 54

2.8 Application to Real Data 56

2.8.1 Results of Real Data Analysis 58

2.9 Discussion 66

2.10 Conclusion 66

Chapter 3 69

3 Rapid Detection of Changes to Individual Modes in Multimodal Power Systems 69

3.1 Introduction 69

3.2 The Stochastic Power System Model Revisited 70

3.3 Application of the Optimal Detection Strategy 71

3.4 Individual Mode Detection Statistic Details 73

3.5 Statistical Characterisation of the Detection Statistic 74

3.6 Results 77

3.6.1 Simulated Results 78

3.7 Real Data Analysis 82

3.8 Verification of Method 85

3.9 Real Data Analysis Results 90

3.10 Discussion 94

3.11 Conclusion 95

Chapter 4 97

4 A Kalman Filtering Approach to Rapidly Detecting Modal Changes 97

4.1 Introduction 97

4.2 Stochastic Power System Model 98

4.3 The Kalman Application in Power System Analysis 101

4.3.1 Kalman formulation 101

4.3.2 State space representation of the power system model 103

4.3.3 Kalman Solution 105

4.3.4 Detection using the Innovation 106

4.4 Simulated Data Results 108

4.4.1 Simulation Type 1- damping change 111

4.4.2 Simulation Type 2- frequency change 112

4.4.3 Simulation Type 3- damping and frequency change 113

4.4.4 Statistics of results 114

4.5 Verification of the Kalman Method 115

4.6 Application to Real Data 116

4.6.1 Part I: Analysis of the Melbourne Data 117

4.6.2 Part II: Combining multi-site data for enhanced SNR and detection ………122

4.7 Guidance in tuning the Kalman Filter 126

4.8 Discussion on real data analysis 127

4.9 Conclusion 128

Chapter 5 129

5 A new class of multi-linear functions for polynomial phase signal analysis 129

5.1 Introduction 129

5.2 The new class of multi-linear functions 132

5.3 Designing new GMFC members 134

5.3.1 Algorithm for estimating the parameters of 4th order PPSs based on 3( )

1 , 3

T n Ω 136

5.4 Derivation of the Asymptotic Mean Squared Errors 138

5.5 Simulations 145

5.6 Application in Power System Monitoring 150

5.6.1 Real Data Analysis and Results 153

5.6.2 Discussion on Real Data Analysis 159

5.7 Conclusion 160

Chapter 6 161

6 Discussion 161

6.1 Comparison of Proposed Detectors 166

Chapter 7 175

7 Conclusions and Future Directions 175

7.1 Conclusion 175

7.2 Future Directions 176

Publications 179

Bibliography 181

Table of Figures

Figure 1-1 States associated with the eastern Australian large

interconnected power system (shaded) State capital cities that represent generation nodes and measurement site location and

ratings are shown 31

Figure 2-1 Model for quasi-continuous modal disturbances in a power system 46

Figure 2-2 Equivalent model for quasi-continuous modal oscillations in a power system 47

Figure 2-3 Energy PDF and Histogram Comparison (60 second window) 52

Figure 2-4 60 second Data Window of Energy Measurements with 1% False Alarm Rate Shown 53

Figure 2-5 Mode Trajectory of QNI Case13 MudpackScript Data 55

Figure 2-6 Output Energy vs 1, 5, 10% thresholds .55

Figure 2-7 Short Term Energy Detection Applied to Real Data, red denotes past data used to formulate long term estimates .57

Figure 2-8 24 hours of recorded angle measurements (2nd October 2004), sites as indicated 59

Figure 2-9 Queensland Estimated Impulse Response .59

Figure 2-10 New South Wales Estimated Impulse Response .60

Figure 2-11 Victorian Estimated Impulse Response 60

Figure 2-12 South Australian Estimated Impulse Response 61

Figure 2-13 Queensland PDF Estimate with white noise verification histogram 61

Figure 2-14 New South Wales PDF Estimate with white noise

verification histogram 62

Figure 2-15 Victorian PDF Estimate with white noise verification histogram 62

Figure 2-16 South Australian PDF Estimate with white noise verification histogram 63

Figure 2-17 Queensland 60 second energy measurements vs various FARs shown 63

Figure 2-18 New South Wales 60 second energy measurements vs various FARs shown 64

Figure 2-19 Victorian 60 second energy measurements vs various FARs shown 64

Figure 2-20 South Australian 60 second energy measurements vs various FARs shown 65

Figure 3-1 Previously introduced stochastic power system model 70

Figure 3-2 Generation of the optimal detection statistic 72

Figure 3-3 Mode 1 test statistic vs alarm threshold 80

Figure 3-4 Mode 2 test statistic vs alarm threshold 81

Figure 3-5 Spectral plot of mode contributions within system frequency response 82

Figure 3-6 Short Term Modal Detection Applied to Real Data 84

Figure 3-7 Case13 Modal Damping and Frequency Trajectory 85

Figure 3-8 Spectral Estimate of Site Magnitude Response 86

Figure 3-9 Estimates of Individual Modal Spectral Contributions - Brisbane (QNI) 87

Figure 3-10 Estimates of Individual Modal Spectral Contributions -

Sydney (NSW) .87

Figure 3-11 Estimates of Individual Modal Spectral Contributions - Adelaide (SA) 88

Figure 3-12 Individual Mode Monitoring - Mode 1 Brisbane .88

Figure 3-13 Individual Mode Monitoring - Mode 1 Sydney .89

Figure 3-14 Individual Mode Monitoring - Mode 1 Adelaide .89

Figure 3-15 Brisbane Mode 1 Test Statistic vs Time (1% FAR) .91

Figure 3-16 Brisbane Mode 2 Test Statistic vs Time (1% FAR) .92

Figure 3-17 Sydney Mode 1 Test Statistic vs Time (1% FAR) .93

Figure 3-18 Sydney Mode 2 Test Statistic vs Time (1% FAR) .93

Figure 3-19 Magnitude spectrum of voltage angles at different sites at 24:00hrs 94

Figure 4-1 Equivalent model for the individual response of a power system to load changes 99

Figure 4-2 General Kalman filter estimator .99

Figure 4-3 Innovation PSD of: a) the 60 second interval prior to the damping change, and b) the 60 second interval subsequent to the damping change .111

Figure 4-4 Innovation PSD of: a) the 60 second interval prior to the frequency shift, and b) the 60 second interval subsequent to the frequency shift 112

Figure 4-5 Innovation PSD of: a) the 60 second interval prior to the damping change, and b) the 60 second interval subsequent to the damping and frequency change 113

Figure 4-6 Analysis of Normalised Innovation prior to sudden

deteriorating damping at 120mins 115

Figure 4-7 Analysis of normalised innovation in the 60 seconds after the deteriorating damping at 121mins 116

Figure 4-8 Melbourne frequency response estimate from LTE at 165 minutes 118

Figure 4-9 Comparison of (a) system output and (b) normalised

Figure 4-14 Normalised (a) Individual PSD at 198-199 mins (b)

Combination PSD at 198-199 mins showing new threshold for CI of 99.9% FAR 125

Figure 5-1 a 4 estimate MSE vs SNR for the final and intermediate

Figure 5-6 b 0 estimate MSE vs SNR for the final parameter estimates.

150

Figure 5-7 Measured Data, (a) Voltage Magnitude and (b) Phase .154

Figure 5-8 Reconstructed Signalz r( )t . 155

Figure 5-9 Example of signal slice used for parameter estimation, down-sampled to 6.25Hz Coloured signal shown is around the first phase disturbance .155

Figure 5-10 Second example of signal slice used for parameter estimation, down-sampled to 6.25Hz Coloured signal shown is around the second phase disturbance .156

Figure 5-11 b 0 estimate .156

Figure 5-12 a 0 estimate (phase deg) 157

Figure 5-13 a 1 estimate, f =2ωπ 157

Figure 5-14 a 2 estimate (frequency rate), ω& .158

Figure 5-15 a 3 estimate, ω&& 158

Figure 5-16 a 4 estimate, ω&&& 159

Figure 6-1 EBD outputs from three sites, NSW, VIC and SA 169

Figure 6-2 OMID Mode 2 outputs from three sites, NSW, VIC and SA 170

Figure 6-3 OMID Mode 1 outputs from three sites, NSW, VIC and SA 171

Figure 6-4 Estimation of Spectral Mode Contributions from three sites, NSW, VIC and SA .172

List of Tables

Table 2-1 Relative Error of Moments 51

Table 2-2 Percentage of Alarms 54

Table 2-3 Percentage of False Alarms over initial 3 hours of Data 65

Table 3-1 Qualitative Reference to Damping Performance (NEMMCO)* .78

Table 3-2 Stationary Modal Parameters and Weights 79

Table 3-3 Damping Changes 80

Table 3-4 Alarms (1% FAR) 81

Table 3-5 Long Term Modal Parameter Estimates 85

Table 3-6 False Alarms (1% FAR) 91

Table 4-1 Qualitative Reference to Damping Performance 109

Table 4-2 Damping and Frequency Changes to Mode 1 110

Table 4-3 Alarms (0.1% FAR) 114

Table 4-4 Damping and Frequency Long Term Estimates over 120-165 mins 118

Table 4-5 SNR Improvement through Combination of Site Analysis 126

Table 5-1 Approximate Formulae for CRLBs (N >> [63] 139 1) Table 5-2 Parameter Values of 4th Order Polyphase Signal used in Simulations 146

Table 6-1 Comparison of detection method test statistics .173

Acknowledgements

The author wishes to thank the following for their support during the period of this research Firstly I’d like to extend a sincere thank you to Associate Professor Peter O’Shea, for his guidance, encouragement, input and patience during the period of this research Dr O’Shea has all the qualities any candidate could ever ask for from a supervisor, both in his technical expertise and his persona, thanks Peter

I would also like to extend a warm thank you to Professor Gerard Ledwich (QUT) for his comments, guidance, input and insightful understanding of the research topic

Other people I’d like to mention and thank are Dr Ed Palmer (QUT) for his contributions, support and friendship over many years and Dr Chaun-

Li Zhang (QUT) for his endless supply of data when I needed it I would also like to thank Maree Farquharson (QUT) who, as fellow candidates, supported and encouraged each other over our respective research periods

In addition I would like to extend a grateful thank you to David Bones (NEMMCO) and David Vowles (University of Adelaide) for giving me permission to use the MudpackScripts as an important validation tool within this thesis, very much appreciated

Finally, I would like to thank the following for the much welcome financial support over the course of the research; Queensland University

of Technology for the APAI and QUTPRA scholarships, the QUT Chancellery for providing the Vice-Chancellor’s Award, the QUT Faculty

of Built Environment and Engineering for the BEE financial top-up and finally my current employer, CEA Technologies Pty Ltd, for providing

me the study leave I required in the final stages of completing this body

of work

Dedication

There are a number of very special people I would like to dedicate this work to; firstly my mother, Virginia Ann Bryan, who with her guidance, support and never ending devotion has helped me mature into the man I

am today I would also like to dedicate this work to my father, Roger Wiltshire, for providing me the qualities that formulate a good engineer Also to my two boys, Peter and Jack, who I am eternally proud of and encouraged by to strive to be a better father and person and finally to three very special and lovely ladies, Catherine Louise Kowalski, Meg Malaika (21/9/1966-19/3/2006) and Leslie Elizabeth Peters who in their own extraordinary way have encouraged, inspired, taught and supported

me in my endeavours over the last few years

Love to you all…

Glossary

AESOPS Analysis of Essentially Spontaneous Oscillations in Power Systems

dB Decibels

Hz Hertz

LTE Long Term Estimator

NEMMCO National Electricity Market Management Company Limited

QLD Queensland

QR Quadratic QUT Queensland University of Technology, Brisbane, Australia

VIC Victoria

W Watts

Monitoring of power system stability is a critical issue for distributed networks with a significant focus on the inter-area oscillations, whereby this stability is largely dependent on all “inter-area oscillations” being positively damped The latter are oscillations that correspond to transient power flows between clusters of generators or plants within a specific area in the large interconnected power system [24] Monitoring and control of these oscillations is vitally important, and has proven far more

difficult than monitoring and control of oscillations associated with a single generator [24]

The inter-area oscillations (or modes) are damped sinusoids, at a given frequency with a relevant damping factor It is the “ring-down” time associated with the damping factor that is of consequence in the transient ability of the system to stabilise post disturbance It is critical that the transient time is short (and stable) to minimise power flows between the

generation clusters and minimise the associated stresses within the

generation/transmission infrastructure As a consequence there has been much work done in the area of damping factor estimation in large distributed power systems Previous estimation methods have employed Eigen analysis [25-27] as well as Prony [28] analysis [29, 30] For accurate damping factor estimation, however, one typically requires large amounts of data [12, 13] Conventional damping factor estimation techniques are therefore not suitable for rapidly detecting sudden modal damping changes This thesis addresses this shortcoming by presenting a variety of new monitoring methods which are able to provide indications

of detrimental modal parameter change with short data records (typically

of the order of minutes)

1.2 The Monitoring of Australia's Large

Interconnected Power System

In Australia, the power system associated with the eastern states is an example of a large interconnected power system The eastern Australian distribution infrastructure contains a number of generation clusters and there are inter-area modal oscillations which arise from the interaction of these clusters A generalised map of the cluster location in eastern Australia is shown in Figure 1-1, with the capital cities representing the generation nodes Also listed in Figure 1-1 are the locations of the GPS

monitoring sites as presented in [22] Due to the importance of

monitoring inter-area oscillations, a number of partners have entered into

collaboration to effect the monitoring These partners include Queensland

University of Technology (QUT), and various transmission distributors as

listed in [22] The wide-area GPS synchronised techniques outlined in

[22] and [23] have provided the real system data analysed in this thesis

Brisbane

Para Adelaide

Rowville Melbourne

Sydney West Sydney

Brisbane

Para Adelaide

Rowville Melbourne

Sydney West Sydney

GPS Measurement Location

& bus rating City

Figure 1-1 States associated with the eastern Australian large interconnected power

system (shaded) State capital cities that represent generation nodes and

measurement site location and ratings are shown

(Template image of Australia sourced from http://www.rrb.com.au/Images/Australia)

1.3 The use of Externally Sourced Simulated Data

for Algorithm Verification

The ultimate goal of power system monitoring algorithms is to perform

reliably in real power system scenarios Before this can be achieved,

however, the algorithms need to be tested via simulations The simulation environment allows conditions to be varied and consequent performance

to be evaluated in a controlled manner For the simulation environment to provide useful testing, however, it must incorporate modelling which is representative of real systems Good modelling strategies help to verify new techniques prior to real data implementation and provide confidence

in real data analysis results The task of creating satisfactory models of large dynamically interconnected systems is a challenging and non-trivial task Fortunately, for the purposes of this PhD research, the author was given access to externally1 created simulation models and data, based on the eastern Australian interconnected power system The modelling of the system was performed by Adelaide Research & Innovation (ARI), a group associated with The University of Adelaide (UA), Australia The formulation of the power system model and associated data was commissioned and contracted by the National Electricity Market Management Company Limited (NEMMCO) to provide benchmark testing of modal estimation methods from various research centres The Centre of Energy and Resource Management within the School of Engineering Systems at Queensland University of Technology was one of the research centres that was benchmark tested in 2004 [31] The

simulated data provided in [31] was referred to as the “MudpackScripts”

by the University of Adelaide authors It consisted of various stationary data sets that were of interest in this thesis The data set of most interest for this thesis is MudpackScript “Case13” which contains large detrimental step changes of damping The details of the changes in the MudpackScripts will be presented in Chapter 2 Section 2.7 where it is first used for technique verification prior to real data analysis

1 Externally in this context means not associated with Queensland University of Technology, Brisbane Australia

1.4 Review of Existing Modal Estimation Methods

With power systems becoming increasingly large and interconnected, the resulting advantages in efficiency have been offset somewhat by the disadvantage of greater vulnerability to system instability This latter problem has made it very important to be able to perform reliable detection of system disturbances from modal oscillation data records These data records can be associated with either a single isolated disturbance or with continuous random disturbances The power industry’s chief concern is in the detection of exponentially growing disturbance modes potentially present in the power system If a growing mode is detected then the power utilities must introduce some dampening

to counteract the mode As a result, methods for fast, reliable and accurate estimation of the modal parameters are very important

1.4.1 Single Isolated Disturbance

The parameter estimation methods in this section of the literature review focus on the power system’s response to a single isolated disturbance

1.4.1.1 Eigenanalysis of Disturbance Modes

There are a number of well established methods that have been used for the analysis of power systems Many of these methods assume that the intrinsically non-linear power system can be approximated as a linear system for small perturbations from the steady state Under this

assumption Kundur et al [32] showed that eigenvalue analysis

techniques could be quite effective Conventionally, eigenanalysis of a power system is carried out by explicitly forming the system matrix, then using the standard QR algorithm to compute the eigenvalues of the matrix [32] Modal oscillation parameters were then obtained from the eigenvalues This basic method has proven to be generally reliable and has been extensively used by power utilities worldwide Unfortunately

this method is unsuitable for large interconnected systems [26] To enable large system mode monitoring the basic Eigen methods above have

undergone various adaptations Byerly et al [7] developed the

best-known algorithm – AESOPS (Analysis of Essentially Spontaneous Oscillations in Power Systems) The advantage of AESOPS is that it does not require the explicit formation of the system state matrix [7] The shortfall of the AESOPS method is the inadequacy for analysing very large interconnected systems

Uchida and Nagao [25] made another development in eigenanalysis by

proposing the use of the “ S matrix method” In this method, it is

assumed that the dynamics of power systems can be linearly

approximated with a set of differential equations of the form, x&= Ax,

where x is the (vector) state of the system and A is the system matrix The S matrix method transforms the matrix, A, into the matrix,

1

number It can be shown that the dominant eigenvalues of S are the same

as the dominant eigenvalues ofA , but with an appropriate choice for h,

can be computed with better numerical precision and speed [25] The refined Lanczos process is also employed to make high-speed calculation

possible [25] Despite the computational advantage of the “ S matrix

method” eigenanalysis has limited application for very large interconnected power systems [33]

1.4.1.2 Spectral Analysis using Prony’s Method

The spectral analysis of modal parameters for power system disturbance monitoring is another area of research which has received much attention

In this approach power system disturbance data records are spectrally analysed immediately after a fault or disturbance One popular technique used for the spectral analysis is Prony’s method, which originated in an earlier century [28] Its ability to be practically implemented, though, was

delayed until the advent of the digital computer Numerical conditioning enhancements developed in 1982 by Kumaresan and Tufts [3] broadened

its applicability to modal estimation C.E Grund et al [33] compared Prony analysis to eigenanalysis and stated that “Prony analysis has an

important advantage over eigenanalysis techniques in that it does not require the derivation of a medium-scale model” Additionally, it can also

be applied to field measurements for the derivation of control design models [33] Many papers have been written on the use of Prony analysis for oscillation modal parameter estimation [34], [29] , [35], with each providing its own insights

It should be noted however that the above applications of Prony’s method assume that the signal contains little noise In practice methods based on the Prony technique are only effective where the noise power is relatively

small Trudnowski et al [30] alluded to this when they stated for Prony analysis “…The accuracy of the mode estimates is limited by the noise

content always found in field measured signals…”

The poor conditioning of Prony’s method exists because of an conditioned matrix inversion in the method To improve the ill conditioning, Kumaresan and Tufts [3] [4] proposed using a “Pseudo-Inverse” matrix, incorporating Singular Value Decomposition (SVD) This technique was further explored by Kumaresan and Tufts in [4] and was an improvement of the backward linear prediction methods proposed

ill-by Nuttall [36], which in turn were improvements of Prony’s original method

This process of applying a truncated SVD analysis effectively increases the SNR in the data prior to obtaining the solution vector In [3], simulations show that this method gives much more accurate estimates of the modal parameters than traditional Prony methods In [2], Kumaresan also provided further enhancement to Prony’s method with the introduction of FIR pre-filtering to reduce the sensitivity of measurement

errors in observed signal samples when determining the parameters of sinusoidal signals

Gomez Martin and Carrion Perez also introduced some extensions in working with noisy data with the application of Prony’s method [6] by using a moving window in both forward and backward directions This application of forward and backward methodologies were further explored by Kannan and Kundu [37]

A time-varying Prony method for instantaneous frequency estimation from low SNR data was introduced by Beex and Shan [38] This was pertinent since power systems do have substantially non-stationary

components on occasions

1.4.1.3 The Sliding Window Derivation

Prony’s method is a parametric spectral analysis method Some authors have pursued solutions using classical spectral analysis based on Fourier methods For example; Poon and Lee [39] developed a technique to determine the modal parameters by employing a Sliding Window Fourier Transform The frequency components of the modes were first identified

in the frequency spectrum The damping constants could then be obtained

by comparing the spectral magnitudes of a given modal component in different time windows These Fourier techniques proved to be quite robust to noise and worked well as long as the oscillation modes were well separated and could be separately distinguished within the Fourier spectral domain

Basically the method developed by Poon and Lee uses the rate of decay

of the Fourier Transform as a rectangular window slides to determine the damping factor of the mode The results of this method provided good correlation compared to conventional techniques However the fundamental limitation of the Poon and Lee method was the tight restrictions on the length of the window that could be used

It was subsequently shown that this restriction was needed to avoid errors due to the interference from the superposition of the positive and negative frequency components [9] This interference was formulated by the large

side lobes of the spectral sinc function introduced by the rectangular

windowing Hence, Poon and Lee specified that the window lengths only

have certain discrete values, at which the interference (zeros in the sinc

function) turned out to be zero The problem was that the window length was dependent upon the modal frequency, and hence this frequency had

to be accurately estimated prior to the windowing implementation Additionally, a different set of windows was required to process every different mode present O’Shea [8] extended Poon and Lee’s Fourier method and showed that a relaxation of the restriction on window lengths could be achieved by applying a smooth tapering window (Kaiser window) rather than a rectangular one [40], [9]

Although the sliding spectral window methods in [39] and [9] were robust

to noise, they only allowed analysis of multiple modes if the modes were sufficiently well separated to be resolved with conventional Fourier techniques To deal with multiple closely spaced modes Poon and Lee [41] developed a modified technique For lightly damped closely spaced low frequency oscillation modes exhibiting beat phenomenon they made use of the imaginary part of the Fourier Transform of the swing curves Their simple dual modal case was modelled by:

resolved using Fourier techniques

Although not specifically stated by Poon and Lee, a major problem with their method was that it was not straightforward to determine either f or 1

2

f It was therefore not straightforward to determine the damping factors

O’Shea [42] showed that a simpler and more reliable method of determining modal parameters for closely spaced modes was to extend the earlier Sliding Window method in [9] by calculating the spectrum in more than two windows Using the results from the multiple spectral windows, a set of simultaneous equations in the desired parameters could

be created These parameters were the complex amplitudes, frequencies and damping factors of the modes These simultaneous equations could then be solved in a least squares sense [4] to obtain estimates for the modal parameters The presented simulations indicated good results

1.4.2 Continuous Random Disturbances

The methods for modal analysis so far discussed all assumed that the data record could be well modelled as a sum of complex exponential modes This is an acceptable model if the record has been obtained after a single isolated disturbance However this is not acceptable for a record obtained from continuous random disturbances (which is the scenario for power systems in normal operation [11]) The following sections investigate modal parameter analysis in relation to continuous random disturbances

1.4.2.1 Autocorrelation Methods

The estimation of modal parameters from data records corresponding to continuous random disturbances was discussed by Ledwich and Palmer in [11] They reasoned that the continuous random disturbances exciting a

power system in normal operation should be fractal in nature, having a 1/f

shaped spectrum [11], i.e it should be equivalent to integrated white noise They also reasoned that a power system could be approximated as

an IIR filter With these assumptions about the excitation and power system, Ledwich and Palmer showed that if one differentiated the output

of the power system, the result would be equivalent to the output of an IIR filter driven by white noise Since the autocorrelation function of a system driven by white noise reveals the impulse response of that system

[43], then the autocorrelation function of the differentiated power system disturbance output should be the impulse response of the power system, i.e it will have the form of a sum of complex exponentials The modal parameters can then be determined using Prony analysis [11]

Autocorrelation techniques were further examined by Banejad and Ledwich [12] to determine resonant frequencies and mode shape by modelling disturbances using white noise to represent customer load variations and an impulse to represent a disturbance The simulation results provided tangible relationships to a known system’s eigenvalue, resonant frequencies and mode shape, but did not make any reference to the limitations of mode spacing

1.4.2.2 Review of Kalman Filter Innovation Strategies

In the 1950s, increased control requirements for advancing avionics led to the formulation of what is now commonly known as the Kalman filter Although earlier radar tracking work by Swerling had formulated very similar algorithms [44] the more highly recognised publications by Kalman [45], [46] then Kalman and Bucy [47] were generally recognised

as the origins of the Kalman filter Since that time the Kalman filter has been recognised as a very important (and optimal) linear estimator; it has been used extensively in a multitude of areas that encompass stochastic models, state and parameter estimation and control requirements There have also been a multitude of Kalman filter variations for non-linear systems, such as the extended Kalman filter [48] and unscented Kalman filter [49] In this thesis, the focus of interest is on the Kalman filter innovation The innovation is defined as the difference between the measured output and the estimated output [50] It is well known that the innovation from a Kalman filter is spectrally white as long as the assumed model parameters are valid [50, 51] However under faulty or changed conditions the innovation sequence will demonstrate large systematic trends as the model will no longer represent the physical system [50]

Various Kalman filter innovation approaches, that target fault detection, diagnosis of dynamic systems and least squares estimation, are presented

in [50-55] In this thesis the Kalman filter model is used to estimate the system output By monitoring the whiteness of the innovation one can detect if there are any sudden detrimental changes in the model parameters [50], otherwise the innovation sequence is equivalent to the original excitation under normal plant conditions [51]

1.5 Review of Frequency Estimation Methods

Although modal damping estimates are of critical importance, the frequency of the modes also provides an opportunity to determine changes in system behaviour and dynamics In the case of large interconnected power systems, if a particular major site disconnects from

a national grid (example South Australia disconnects from the Australian Eastern network) then the resulting power system will undergo a dynamic shift in an attempt to re-establish an equilibrium state In doing so it is expected that the frequencies of the remaining modes will also change Therefore to rapidly detect and estimate the changes in modal frequencies

is of critical importance

The nature of the frequency changes that occur in a power system over time will not be known precisely To allow for the arbitrary nature of the frequency trajectories, polynomial modelling will be used Note that if the frequency trajectory is a polynomial as a function of time, then the phase trajectory will also be polynomial It will be assumed that the component/mode in question is given by:

where b 0 is the amplitude, the polynomial phase coefficients are given by

{a a0, 1,K,a P} and the phase is: